دانلود کتاب Advanced Calculus

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PREFACE
CONTENTS
Chapter I THE REAL NUMBER SYSTEM
101. Introduction
102. Axioms of a field
103. Exercises
104. Axioms of an ordered field
105. Exercises
106. Positive integers and mathematical induction
107. Exercises
108. Integers and rational numbers
109. Exercises
110. Geometrical representation and absolute value
111. Exercises
112. Axiom of completeness
113. Consequences of completeness
114. Exercises
Chapter 2 FUNCTIONS, SEQUENCES, LIMITS, CONTINUITY
201. Functions and sequences
202. Limit of a sequence
203. Exercises
204. Limit theorems for sequences
205. Exercises
206. Limits of functions
207. Limit theorems for functions
208. Exercises
209. Continuity
210. Types of discontinuity
211. Continuity theorems
212. Exercises
213. More theorems on continuous functions
214. Existence of √2 and other roots
215. Monotonic functions and their inverses
216. Exercises
*217. A fundamental theorem on bounded sequences
*218. Proofs of some theorems on continuous functions
*219. The Cauchy criterion for convergence of a sequence
*220. Exercises
*221. Sequential criteria for continuity and existence of limits
*222. The Cauchy criterion for functions
*223. Exercises
*224. Uniform continuity
*225. Exercises
Chapter 3 DIFFERENTIATION
301. Introduction
302. The derivative
303. One-sided derivatives
304. Exercises
305. Rolle\'s theorem and the Law of the Mean
306. Consequences of the Law of the Mean
307. The Extended Law of the Mean
308. Exercises
309. Maxima and minima
310. Exercises
311. Differentials
312. Approximations by differentials
313. Exercises
314. L\'Hospital\'s Rule. Introduction
315. The indeterminate form 0/0
316. The indeterminate form ∞/∞
317. Other indeterminate forms
318. Exercises
319. Curve tracing
320. Exercises
*321. Without loss of generality
*322. Exercises
Chapter 4 INTEGRATION
401. The definite integral
402. Exercises
*403. More integration theorems
*404. Exercises
405. The Fundamental Theorem of Integral Calculus
406. Integration by substitution
407. Exercises
408. Sectional continuity and smoothness
409. Exercises
410. Reduction formulas
411. Exercises
412. Improper integrals, introduction
413. Improper integrals, finite interval
414. Improper integrals, infinite interval
415. Comparison tests. Dominance
416. Exercises
*417. The Riemann-Stieltjes integral
*418. Exercises
Chapter 5 SOME ELEMENTARY FUNCTIONS
*501. The exponential and logarithmic functions
*502. Exercises
*503. The trigonometric functions
*504. Exercises
505. Some integration formulas
506. Exercises
507. Hyperbolic functions
508. Inverse hyperbolic functions
509. Exercises
*510. Classification of numbers and functions
*511. The elementary functions
*512. Exercises
Chapter 6 FUNCTIONS OF SEVERAL VARIABLES
601. Introduction
602. Neighborhoods in the Euclidean plane
603. Point sets in the Euclidean plane
604. Sets in higher-dimensional Euclidean spaces
605. Exercises
606. Functions and limits
607. Iterated limits
608. Continuity
609. Limit and continuity theorems
610. More theorems on continuous functions
611. Exercises
612. More general functions. Mappings
*613. Sequences of points
*614. Point sets and sequences
*615. Compactness and continuity
*616. Proofs of two theorems
*617. Uniform continuity
618. Exercises
Chapter 7 SOLID ANALYTIC GEOMETRY AND VECTORS
701. Introduction
702. Vectors and scalars
703. Addition and subtraction of vectors. Magnitude
704. Linear combinations of vectors
705. Exercises
706. Direction angles and cosines
707. The scalar or inner or dot product
708. Vectors orthogonal to two vectors
709. Exercises
710. Planes
711. Lines
712. Exercises
713. Surfaces. Sections, traces, intercepts
714. Spheres
715. Cylinders
716. Surfaces of revolution
717. Exercises
718. The standard quadric surfaces
719. Exercises
Chapter 8 ARCS AND CURVES
801. Duhamel\'s principle for integrals
*802. A proof with continuity hypotheses
803. Arcs and curves
804. Arc length
805. Integral form for arc length
*806. Remark concerning the trigonometric functions
807. Exercises
808. Cylindrical and spherical coordinates
809. Arc length in rectangular, cylindrical, and spherical coordinates
810. Exercises
811. Curvature and radius of curvature in two dimensions
812. Circle of curvature
*813. Evolutes and involutes
814. Exercises
Chapter 9 PARTIAL DIFFERENTIATION
901. Partial derivatives
902. Partial derivatives of higher order
*903. Equality of mixed partial derivatives
904. Exercises
905. The fundamental increment formula
906. Differentials
907. Change of variables. The chain rule
*908. Homogeneous functions. Euler\'s theorem
909. Exercises
*910. Directional derivatives. Tangents and normals
*911. Exercises
912. The Law of the Mean
913. Approximations by differentials
914. Maxima and minima
915. Exercises
916. Differentiation of an implicit function
917. Some notational pitfalls
918. Exercises
919. Envelope of a family of plane curves
920. Exercises
921. Several functions defined implicitly. Jacobians
922. Coordinate transformations. Inverse transformations
923. Functional dependence
924. Exercises
925. Extrema with one constraint. Two variables
926. Extrema with one constraint. More than two variables
927. Extrema with more than one constraint
928. Lagrange multipliers
929. Exercises
*930. Differentiation under the integral sign. Leibnitz\'s rule
*931. Exercises
*932. The Implicit Function Theorem
*933. Existence theorem for inverse transformations
*934. Sufficiency conditions for functional dependence
*935. Exercises
Chapter 10 MULTIPLE INTEGRALS
1001. Introduction
1002. Double integrals
1003. Area
1004. Second formulation of the double integral
*1005. Inner and outer area. Criterion for area
*1006. Theorems on double integrals
*1007. Proof of the second formulation
1008. Iterated integrals, two variables
*1009. Proof of the Fundamental Theorem
1010. Exercises
1011. Triple integrals. Volume
1012. Exercises
1013. Double integrals in polar coordinates
1014. Volumes with double integrals in polar coordinates
1015. Exercises
1016. Mass of a plane region of variable density
1017. Moments and centroid of a plane region
1018. Exercises
1019. Triple integrals, cylindrical coordinates
1020. Triple integrals, spherical coordinates
1021. Mass, moments, and centroid of a space region
1022. Exercises
1023. Mass, moments, and centroid of an arc
1024. Attraction
1025. Exercises
1026. Jacobians and transformations of multiple integrals
1027. General discussion
1028. Exercises
Chapter 11 INFINITE SERIES OF CONSTANTS
1101. Basic definitions
1102. Three elementary theorems
1103. A necessary condition for convergence
1104. The geometric series
1105. Positive series
1106. The integral test
1107. Exercises
1108. Comparison tests. Dominance
1109. The ratio test
1110. The root test
1111. Exercises
*1112. More refined tests
*1113. Exercises
1114. Series of arbitrary terms
1115. Alternating series
1116. Absolute and conditional convergence
1117. Exercises
1118. Groupings and rearrangements
1119. Addition, subtraction, and multiplication of series
*1120. Some aids to computation
1121. Exercises
Chapter 12 POWER SERIES
1201. Interval of convergence
1202. Exercises
1203. Taylor series
1204. Taylor\'s formula with a remainder
1205. Expansions of functions
1206. Exercises
1207. Some Maclaurin series
1208. Elementary operations with power series
1209. Substitution of power series
1210. Integration and differentiation of power series
1211. Exercises
1212. Indeterminate expressions
1213. Computations
1214. Exercises
1215. Taylor series, several variables
1216. Exercises
*Chapter 13 UNIFORM CONVERGENCE AND LIMITS
*1301. Uniform convergence of sequences
*1302. Uniform convergence of series
*1303. Dominance and the Weierstrass M-test
*1304. Exercises
*1305. Uniform convergence and continuity
*1306. Uniform convergence and integration
*1307. Uniform convergence and differentiation
*1308. Exercises
*1309. Power series. Abel\'s theorem
*1310. Proof of Abel\'s theorem
*1311. Exercises
*1312. Functions defined by power series. Exercises
*1313. Uniform limits of functions
*1314. Three theorems on uniform limits
*1315. Exercises
*Chapter 14 IMPROPER INTEGRALS
*1401. Introduction. Review
*1402. Alternating integrals. Abel\'s test
*1403. Exercises
*1404. Uniform convergence
*1405. Dominance and the Weierstrass M-test
*1406. The Cauchy criterion and Abel\'s test for uniform convergence
*1407. Three theorems on uniform convergence
*1408. Evaluation of improper integrals
*1409. Exercises
*1410. The gamma function
*1411. The beta function
*1412. Exercises
*1413. Infinite products
*1414. Wallis\'s infinite product for π
*1415. Euler\'s constant
*1416. Stirling\'s formula
*1417. Weierstrass\'s infinite product for 1/Γ(α)
*1418. Exercises
*1419. Improper multiple integrals
*1420. Exercises
Chapter 15 COMPLEX VARIABLES
1501. Introduction
1502. Complex numbers
1503. Embedding of the real numbers
1504. The number i
1505. Geometrical representation
1506. Polar form
1507. Conjugates
1508. Roots
1509. Exercises
1510. Limits and continuity
1511. Sequences and series
1512. Exercises
1513. Complex-valued functions of a real variable
1514. Exercises
*1515. The Fundamental Theorem of Algebra
Chapter 16 FOURIER SERIES
1601. Introduction
1602. Linear function spaces
1603. Periodic functions. The space R_2π
1604. Inner product. Orthogonality. Distance
1605. Least squares. Fourier coefficients
1606. Fourier series
1607. Exercises
1608. A convergence theorem. The space S_2π
1609. Bessel\'s inequality. Parseval\'s equation
1610. Cosine series. Sine series
1611. Other intervals
1612. Exercises
*1613. Partial sums of Fourier series
*1614. Functions with one-sided limits
*1615. The Riemann-Lebesgue Theorem
*1616. Proof of the convergence theorem
*1617. Fejer\'s summability theorem
*1618. Uniform summability
*1619. Weierstrass\'s theorem
*1620. Density of trigonometric polynomials
*1621. Some consequences of density
*1622. Further remarks
*1623. Other orthonormal systems
*1624. Exercises
1625. Applications of Fourier series. The vibrating string
1626. A heat conduction problem
1627. Exercises
Chapter 17 VECTOR ANALYSIS
1701. Introduction
1702. The vector or outer or cross product
1703. The triple scalar product. Orientation in space
1704. The triple vector product
1705. Exercises
1706. Coordinate transformations
1707. Translations
1708. Rotations
1709. Exercises
1710. Scalar and vector fields. Vector functions
1711. Ordinary derivatives of vector functions
1712. The gradient of a scalar field
1713. The divergence and curl of a vector field
1714. Relations among vector operations
1715. Exercises
*1716. Independence of the coordinate system
*1717. Curvilinear coordinates. Orthogonal coordinates
*1718. Vector operations in orthogonal coordinates
*1719. Exercises
Chapter 18 LINE AND SURFACE INTEGRALS
1801. Introduction
1802. Line integrals in the plane
1803. Independence of path and exact differentials
1804. Exercises
1805. Green\'s Theorem in the plane
1806. Local exactness
1807. Simply- and multiply-connected regions
1808. Equivalences in simply-connected regions
1809. Exercises
*1810. Analytic functions of a complex variable. Exercises
1811. Surface elements
1812. Smooth surfaces
1813. Schwarz\'s example
1814. Surface area
1815. Exercises
1816. Surface integrals
1817. Orientable smooth surfaces
1818. Surfaces with edges and corners
1819. The divergence theorem
1820. Green\'s identities
1821. Harmonic functions
1822. Exercises
1823. Orientable sectionally smooth surfaces
1824. Stokes\'s Theorem
1825. Independence of path. Scalar potential
*1826. Vector potential
1827. Exercises
*1828. Exterior differential forms. Exercises
Chapter 19 DIFFERENTIAL GEOMETRY
1901. Introduction
1902. Curvature. Osculating plane
1903. Applications to kinematics
1904. Torsion. The Frenet formulas
1905. Local behavior
1906. Exercises
1907. Curves on a surface. First fundamental form
1908. Intersections of smooth surfaces
1909. Plane sections. Meusnier\'s theorem
1910. Normal sections. Mean and total curvature
1911. Second fundamental form
1912. Exercises
ANSWERS TO PROBLEMS
INDEX